If `x^(y)=y^(x)` show that `(x/y)^(x/y)=x^((x)/(y)-1)` if further x=2y then prove that y=2

If x^y=y^x , then prove that (x/y)^(x/y)=(x)^((x)/(y)-1) .

If (x-y)*e^((x)/(x-y))=a , then prove that y(dy)/(dx)+x=2y .

If x^(y).y^(x)=1, prove that (dy)/(dx)=-(y(y+x log y))/(x(y log x+x))

If x^(y)y^(x),=1, prove that (dy)/(dx),=-(y(y+x log y))/(x(y log x+x))

If x+y=e^(x-y) then show that (d^(2)y)/(dx^(2))=(4(x+y))/((1+x+y)^(3))

y=(x)/(1+(x)/(2+(x)/(1+(x)/(2)+...))) then prove that y'=(1)/(1+y)

If y=x+(1)/(x+(1)/(x+…oo)) , prove that (dy)/(dx)=(y)/(2y-x) .

If x ^(y) + y ^(x) = a ^(b) then show that (dy)/(dx) =- ((y x ^(y-1) + y^(x) log y)/( x ^(y) log x + x y ^(x -1)))

If S_(n)=(x+y)+(x^(2)+xy+y^(2))+(x^(3)+x^(2)y+y^(2)x+y^(3))+…n terms then prove that (x-y)S_(n)=[(x^(2)(x^(n)-1))/(x-1)-(y^(2)y^(n)-1)/(y-1)] .

If (x-y)x^((x)/(x-y))=a, prove that y(dy)/(dx)+x=2y